total relation

A (binary) relation \sim on a set AA is total if any two elements are related in one order or the other:

(x,y:A),xyyx\forall (x, y: A),\; x \sim y \;\vee\; y \sim x

In the language of the 22-poset-with-duals Rel of sets and relations, a relation R:AAR: A \to A is total if its union with its reverse is the universal relation:

A×ARR opA \times A \subseteq R \cup R^{op}

Of course, this containment is in fact an equality.

A total relation is necessarily reflexive.

Note that an entire relation is sometimes called ‘total’, but these are unrelated concepts. The ‘total’ there is in the sense of a total (as opposed to partial) function, while the ‘total’ here is in the sense of total (as opposed to partial) order.

Revised on December 16, 2016 07:13:45 by Toby Bartels (